# Śleszyński–Pringsheim theorem

In mathematics, the Śleszyński–Pringsheim theorem is a statement about convergence of certain continued fractions. It was discovered by Ivan Śleszyński[1] and Alfred Pringsheim[2] in the late 19th century.[3]

It states that if an, bn, for n = 1, 2, 3, ... are real numbers and |bn| ≥ |an| + 1 for all n, then

$\cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+ \ddots}}}$

converges absolutely to a number ƒ satisfying 0 < |ƒ| < 1,[4] meaning that the series

$f = \sum_n \left\{ \frac{A_n}{B_n} - \frac{A_{n-1}}{B_{n-1}}\right\},$

where An / Bn are the convergents of the continued fraction, converges absolutely.

## Notes and references

1. ^ Слешинскій, И. В. (1889). "Дополненiе къ замѣткѣ о сходимости непрерывныхъ дробей". Матем. сб. (in Russian) 14 (3): 436–438.
2. ^ Pringsheim, A. (1898). "Ueber die Convergenz unendlicher Kettenbrüche". Münch. Ber. (in German) 28: 295–324. JFM 29.0178.02.
3. ^ W.J.Thron has found evidence that Pringsheim was aware of the work of Śleszyński before he published his article; see Thron, W. J. (1992). "Should the Pringsheim criterion be renamed the Śleszyński criterion?". Comm. Anal. Theory Contin. Fractions 1: 13–20. MR 1192192.
4. ^ Lorentzen, L.; Waadeland, H. (2008). Continued Fractions: Convergence theory. Atlantic Press. p. 129.